Download - Strut and Tie
![Page 1: Strut and Tie](https://reader031.vdocument.in/reader031/viewer/2022012403/546a5003b4af9f4c578b4686/html5/thumbnails/1.jpg)
8.1
Strut-and-Tie Model
• Background• AASHTO LRFD Provisions• Design Example
![Page 2: Strut and Tie](https://reader031.vdocument.in/reader031/viewer/2022012403/546a5003b4af9f4c578b4686/html5/thumbnails/2.jpg)
8.2Background
STM is a Truss AnalogyTruss Analogy Used in Standard and LRFD SpecificationsVn = Vc + Vs Vs = [Asfy/s]d(cotθ)- AASHTO Standard
Vs 45º Truss- AASHTO LRFD
Vs Variable Angle Truss
![Page 3: Strut and Tie](https://reader031.vdocument.in/reader031/viewer/2022012403/546a5003b4af9f4c578b4686/html5/thumbnails/3.jpg)
8.3
![Page 4: Strut and Tie](https://reader031.vdocument.in/reader031/viewer/2022012403/546a5003b4af9f4c578b4686/html5/thumbnails/4.jpg)
8.4STM in Codes
CSA 23.3-84OHBDC Third Edition, 1991AASHTO LRFD - First Edition, 1994CHBDC - 2000ACI 318-02 Appendix A
![Page 5: Strut and Tie](https://reader031.vdocument.in/reader031/viewer/2022012403/546a5003b4af9f4c578b4686/html5/thumbnails/5.jpg)
8.5Quiz
A Three-Span Concrete Beam Is Built Monolithically, with Continuous Reinforcement Placed Only in the Bottom of the BeamHow Will this Beam Perform Under Service Loads? and at Ultimate?
![Page 6: Strut and Tie](https://reader031.vdocument.in/reader031/viewer/2022012403/546a5003b4af9f4c578b4686/html5/thumbnails/6.jpg)
8.6As Built
![Page 7: Strut and Tie](https://reader031.vdocument.in/reader031/viewer/2022012403/546a5003b4af9f4c578b4686/html5/thumbnails/7.jpg)
8.7Under Service Loads- Uncracked Condition -
![Page 8: Strut and Tie](https://reader031.vdocument.in/reader031/viewer/2022012403/546a5003b4af9f4c578b4686/html5/thumbnails/8.jpg)
8.8Under Service Loads- Cracked Condition -
![Page 9: Strut and Tie](https://reader031.vdocument.in/reader031/viewer/2022012403/546a5003b4af9f4c578b4686/html5/thumbnails/9.jpg)
8.9Observations
Reinforcement Becomes Active After Concrete CracksRedistribution of Internal Stresses Occurs After Concrete CracksAfter Cracking, Concrete Structures Behave the Way they Are ReinforcedFor Best Serviceability, the Reinforcement Must Follow the Flow of Elastic Tensile Stresses
![Page 10: Strut and Tie](https://reader031.vdocument.in/reader031/viewer/2022012403/546a5003b4af9f4c578b4686/html5/thumbnails/10.jpg)
8.10
Strut-and-Tie Model (STM)
Valuable tool for the analysis and design of concrete members, especially for regions where the plane sections assumption of beam theory does not apply
![Page 11: Strut and Tie](https://reader031.vdocument.in/reader031/viewer/2022012403/546a5003b4af9f4c578b4686/html5/thumbnails/11.jpg)
8.11Deep Beam Stress Trajectories
![Page 12: Strut and Tie](https://reader031.vdocument.in/reader031/viewer/2022012403/546a5003b4af9f4c578b4686/html5/thumbnails/12.jpg)
8.12STM for D-Regions
Tee Beam
Dapped Beam
![Page 13: Strut and Tie](https://reader031.vdocument.in/reader031/viewer/2022012403/546a5003b4af9f4c578b4686/html5/thumbnails/13.jpg)
8.13Past Practice
D-Regions Designed Based On:» Experience» Empirical Rules» Rules of Thumb
![Page 14: Strut and Tie](https://reader031.vdocument.in/reader031/viewer/2022012403/546a5003b4af9f4c578b4686/html5/thumbnails/14.jpg)
8.14Basic Description of theStrut-and-Tie Model
A design tool for “disturbed” regions where the flow of stresses is non-uniform and the usual rules of analysis do not applyA rational approach to visualize the flow of forces at the strength limit state based on the variable-angle truss analogyA unified approach that considers all load effects simultaneouslyA highly flexible and conceptual method that recognizes that several possible solutions may exist for any problem
![Page 15: Strut and Tie](https://reader031.vdocument.in/reader031/viewer/2022012403/546a5003b4af9f4c578b4686/html5/thumbnails/15.jpg)
8.15STM Basic Principle
Concrete is Strong in CompressionCompression Struts
Steel is Strong in TensionTension Ties
![Page 16: Strut and Tie](https://reader031.vdocument.in/reader031/viewer/2022012403/546a5003b4af9f4c578b4686/html5/thumbnails/16.jpg)
8.16
P2
φ >
P2
P
![Page 17: Strut and Tie](https://reader031.vdocument.in/reader031/viewer/2022012403/546a5003b4af9f4c578b4686/html5/thumbnails/17.jpg)
8.17
NodalZonesP
2
P
P2
CC
T T
C CStrut
Fill
Fill
Tie
Fill
![Page 18: Strut and Tie](https://reader031.vdocument.in/reader031/viewer/2022012403/546a5003b4af9f4c578b4686/html5/thumbnails/18.jpg)
8.18
T
C
T
C
C C
P
P2
φ > A f Ts y
P2
φ >
Af
Cc
cu
φ > A f Ts y
φ >
Af
Cc
cu
![Page 19: Strut and Tie](https://reader031.vdocument.in/reader031/viewer/2022012403/546a5003b4af9f4c578b4686/html5/thumbnails/19.jpg)
8.19Basic Concepts
Visualize a truss-like system to transfer load to the supports where:• Compressive forces are resisted by
concrete “struts”• Tensile forces are resisted by steel
“ties”• Struts and ties meet at “nodes”
For best serviceability, the model should follow the elastic flow of forces
![Page 20: Strut and Tie](https://reader031.vdocument.in/reader031/viewer/2022012403/546a5003b4af9f4c578b4686/html5/thumbnails/20.jpg)
8.20Strut-and-Tie Model for Simple Span Beam
![Page 21: Strut and Tie](https://reader031.vdocument.in/reader031/viewer/2022012403/546a5003b4af9f4c578b4686/html5/thumbnails/21.jpg)
8.21Examples of Strut-and-Tie Models
![Page 22: Strut and Tie](https://reader031.vdocument.in/reader031/viewer/2022012403/546a5003b4af9f4c578b4686/html5/thumbnails/22.jpg)
8.22Methods for Formulating Strut-and-Tie Models
Stress trajectories from elastic analysis
Load path approach
Experimentally
Standard models
![Page 23: Strut and Tie](https://reader031.vdocument.in/reader031/viewer/2022012403/546a5003b4af9f4c578b4686/html5/thumbnails/23.jpg)
8.23Deep Beam Stress Trajectories
![Page 24: Strut and Tie](https://reader031.vdocument.in/reader031/viewer/2022012403/546a5003b4af9f4c578b4686/html5/thumbnails/24.jpg)
8.24Examples of Strut-and-Tie Models
![Page 25: Strut and Tie](https://reader031.vdocument.in/reader031/viewer/2022012403/546a5003b4af9f4c578b4686/html5/thumbnails/25.jpg)
8.25Examples of Strut-and-Tie Models
![Page 26: Strut and Tie](https://reader031.vdocument.in/reader031/viewer/2022012403/546a5003b4af9f4c578b4686/html5/thumbnails/26.jpg)
8.26
Procedures for Load Path Approach
Find reactionsSubdivide loads and internal forces- Replace stresses with resultants- Replace asymmetrical stresses with
couple and resultantProvide struts and ties to provide load pathLocate ties using practical dimensions
![Page 27: Strut and Tie](https://reader031.vdocument.in/reader031/viewer/2022012403/546a5003b4af9f4c578b4686/html5/thumbnails/27.jpg)
8.27STM from Tests - Dapped Beam
![Page 28: Strut and Tie](https://reader031.vdocument.in/reader031/viewer/2022012403/546a5003b4af9f4c578b4686/html5/thumbnails/28.jpg)
8.28Dapped Beam
![Page 29: Strut and Tie](https://reader031.vdocument.in/reader031/viewer/2022012403/546a5003b4af9f4c578b4686/html5/thumbnails/29.jpg)
8.29Types of Nodes(Schlaich et al. 1987)
C - CompressionT - Tension
TTT
CTT
CCT
CCC
![Page 30: Strut and Tie](https://reader031.vdocument.in/reader031/viewer/2022012403/546a5003b4af9f4c578b4686/html5/thumbnails/30.jpg)
8.30AssumptionsTies yield before struts crush (for ductility)Reinforcement adequately anchoredForces in struts and ties are uniaxialTension in concrete is neglectedExternal forces applied at nodesPrestressing is a load
Equilibrium must be maintained
![Page 31: Strut and Tie](https://reader031.vdocument.in/reader031/viewer/2022012403/546a5003b4af9f4c578b4686/html5/thumbnails/31.jpg)
8.31Strut-and-Tie Model Design Procedure
![Page 32: Strut and Tie](https://reader031.vdocument.in/reader031/viewer/2022012403/546a5003b4af9f4c578b4686/html5/thumbnails/32.jpg)
8.32Examples of Good and Poor Strut-and-Tie Models
![Page 33: Strut and Tie](https://reader031.vdocument.in/reader031/viewer/2022012403/546a5003b4af9f4c578b4686/html5/thumbnails/33.jpg)
8.33Factors Affecting Size of Strut
Width of the strut is affected by:• Location and distribution of reinforcement (tie)
and its anchorage• Size and location of bearing
![Page 34: Strut and Tie](https://reader031.vdocument.in/reader031/viewer/2022012403/546a5003b4af9f4c578b4686/html5/thumbnails/34.jpg)
8.34Strut-and-Tie vs. Traditional Analysis/Design
Traditional section analysis/designLinear strain over member depthUniform shear stress distributionNot valid for D-regions
Strut-and-tieRegions with nonlinear strain distribution» Deep beams, pile caps» Brackets, beam ledges, P/T anchors» Shear span/member height < 2
![Page 35: Strut and Tie](https://reader031.vdocument.in/reader031/viewer/2022012403/546a5003b4af9f4c578b4686/html5/thumbnails/35.jpg)
8.35
a/d
V/bdfc’
Source: Prestressed Concrete Structures by Collins & Mitchell
![Page 36: Strut and Tie](https://reader031.vdocument.in/reader031/viewer/2022012403/546a5003b4af9f4c578b4686/html5/thumbnails/36.jpg)
8.36LRFD 5.2 - Definitions
Strut-and-Tie Model - A model used principally in regions of concentrated forces and geometric discontinuities to determine concrete proportions and reinforcement quantities and patterns based on assumed compression struts in the concrete, tensile ties in the reinforcement, and the geometry of nodes at their points of intersection
![Page 37: Strut and Tie](https://reader031.vdocument.in/reader031/viewer/2022012403/546a5003b4af9f4c578b4686/html5/thumbnails/37.jpg)
8.375.6.3.1 D-RegionsStrut-and-tie models may be used to
determine internal force effects near supportsand the points of application of concentrated loads at strength and extreme event limit states.
The strut-and-tie model should be considered for the design of deep footings and pile caps or other situations in which the distance between the centers of applied load and the supporting reactions is less than about twice the member thickness.
![Page 38: Strut and Tie](https://reader031.vdocument.in/reader031/viewer/2022012403/546a5003b4af9f4c578b4686/html5/thumbnails/38.jpg)
8.385.8.1.1 D-RegionsComponents in which the distance from
the point of zero shear to the face of the support is less than 2d, or components for which a load causing more than ½ of the shear at a support is closer than 2d from the face of the support, may be considered to be deep components for which the provisions of Article 5.6.3 and the detailing requirements of Article 5.13.2.3 apply.
![Page 39: Strut and Tie](https://reader031.vdocument.in/reader031/viewer/2022012403/546a5003b4af9f4c578b4686/html5/thumbnails/39.jpg)
8.39Strength Limit State for STM
Pr = ϕ Pn (5.6.3.2-1)
where:
Pr = Factored resistance
Pn = Nominal resistance of strut or tie
ϕ = Resistance factor for tension or compression (5.5.4.2)
![Page 40: Strut and Tie](https://reader031.vdocument.in/reader031/viewer/2022012403/546a5003b4af9f4c578b4686/html5/thumbnails/40.jpg)
8.40
LRFD 5.6.3.3Unreinforced strut:
Pn = fcu Acs (5.6.3.3.1-1)
Reinforced strut:
Pn = fcu Acs + fy Ass (5.6.3.3.4-1)
where:ϕ = 0.70 for compression in strut-and-tie models
(LRFD 5.5.4.2.1)Acs= effective cross-sectional area of strut
(LRFD 5.6.3.3.2)Ass= area of reinforcement in the strut
Strength of Struts
![Page 41: Strut and Tie](https://reader031.vdocument.in/reader031/viewer/2022012403/546a5003b4af9f4c578b4686/html5/thumbnails/41.jpg)
8.41STM for Deep BeamLRFD Fig. C5.6.3.2-1
![Page 42: Strut and Tie](https://reader031.vdocument.in/reader031/viewer/2022012403/546a5003b4af9f4c578b4686/html5/thumbnails/42.jpg)
8.42
LRFD 5.6.3.3.2
Determined by considering available concrete area and anchorage conditions.
When anchored by reinforcement, strut may extend from the anchored bar.
C-T-T Nodea) Strut Anchored by Reinforcement
Effective Cross-Sectional Area of Strut, Acs
![Page 43: Strut and Tie](https://reader031.vdocument.in/reader031/viewer/2022012403/546a5003b4af9f4c578b4686/html5/thumbnails/43.jpg)
8.43Effective Cross-Sectional Area of Strut, AcsLRFD 5.6.3.3.2
C-C-T Nodeb) Strut Anchored by Bearing and Reinforcement
![Page 44: Strut and Tie](https://reader031.vdocument.in/reader031/viewer/2022012403/546a5003b4af9f4c578b4686/html5/thumbnails/44.jpg)
8.44Effective Cross-Sectional Area of Strut, AcsLRFD 5.6.3.3.2
C-C-C Nodec) Strut Anchored by Bearing and Strut
![Page 45: Strut and Tie](https://reader031.vdocument.in/reader031/viewer/2022012403/546a5003b4af9f4c578b4686/html5/thumbnails/45.jpg)
8.45Limiting Compressive Stress in Strut
LRFD 5.6.3.3.3
where:
( )
(IN/IN) tie tension the of direction the in
concrete the in strain tensile the
(DEG) ties tension
adjoining and strut ecompressiv
the between angle smallest the
stress ecompressiv limiting thef
cot0.002
f85.170 0.8
ff
s
s
cu
s2
ss1
c1
ccu
=
==
++=
′≤+′
=
ε
α
αεεε
ε0
![Page 46: Strut and Tie](https://reader031.vdocument.in/reader031/viewer/2022012403/546a5003b4af9f4c578b4686/html5/thumbnails/46.jpg)
8.46Strength of TieLRFD 5.6.3.4.1
Pn = Ast fy + Aps ( fpe + fy )
where
Ast = Total area of longitudinal mild steel reinforcement on the tie
Aps = Area of prestressing steelfy = Yield strength of mild steel longitudinal
reinforcementfpe = Stress in prestressing steel due to prestress after
losses
![Page 47: Strut and Tie](https://reader031.vdocument.in/reader031/viewer/2022012403/546a5003b4af9f4c578b4686/html5/thumbnails/47.jpg)
8.47Development of Ties
If x < ld fs = fy (x/ld)
Critical Section
= x
![Page 48: Strut and Tie](https://reader031.vdocument.in/reader031/viewer/2022012403/546a5003b4af9f4c578b4686/html5/thumbnails/48.jpg)
8.48Development of Ties (ACI 318)
![Page 49: Strut and Tie](https://reader031.vdocument.in/reader031/viewer/2022012403/546a5003b4af9f4c578b4686/html5/thumbnails/49.jpg)
8.49
Element Limiting Stress ϕ
1 - CCC Node 0.85 fc’ 0.70
2 - CCT Node 0.75fc’ 0.70
3 - CTT or TTT Node 0.65fc’ 0.70
4 - Strut fcu 0.70
5 - Tie fy or (fpe + fy) 0.90 or 1.00
Limiting Stresses for STM ElementsLRFD 5.6.3.3 - 5.6.3.5
![Page 50: Strut and Tie](https://reader031.vdocument.in/reader031/viewer/2022012403/546a5003b4af9f4c578b4686/html5/thumbnails/50.jpg)
8.50Crack Control Reinforcement
LRFD 5.6.3.6
Provide orthogonal grid of reinforcement near each face of D-RegionMaximum Bar Spacing = 12 in.Ratio As / Ag ≥ 0.003 in each of the orthogonal directionsCrack control reinforcement, located within tie, considered as part of tie
![Page 51: Strut and Tie](https://reader031.vdocument.in/reader031/viewer/2022012403/546a5003b4af9f4c578b4686/html5/thumbnails/51.jpg)
8.51Summary
1. Visualize flow of stresses2. Sketch an idealized strut-and-tie model3. Select area of ties4. Check nodal zone stresses5. Check strength of struts6. Provide adequate anchorage for ties
![Page 52: Strut and Tie](https://reader031.vdocument.in/reader031/viewer/2022012403/546a5003b4af9f4c578b4686/html5/thumbnails/52.jpg)
8.52
![Page 53: Strut and Tie](https://reader031.vdocument.in/reader031/viewer/2022012403/546a5003b4af9f4c578b4686/html5/thumbnails/53.jpg)
8.53Strut-and-Tie Model
![Page 54: Strut and Tie](https://reader031.vdocument.in/reader031/viewer/2022012403/546a5003b4af9f4c578b4686/html5/thumbnails/54.jpg)
8.54Strut-and-Tie Model
![Page 55: Strut and Tie](https://reader031.vdocument.in/reader031/viewer/2022012403/546a5003b4af9f4c578b4686/html5/thumbnails/55.jpg)
8.55Design Examples
1. Two Column Bent Cap2. Spread Footing3. Pile Cap4. Dapped-End Beam5. Hammerhead Pier